Some Properties of Transforms in Cultural Theory

Abstract

It is shown that, in certain circumstances, systems of cultural rules may be represented by doubly stochastic matrices denoted , called "possibility transforms," and by certain real valued "possibility densities" π = (π 1 , π 2 , . . . , π n ) with inner product π, π = i π 2 i = 1. We may characterize a certain problem of ethnographic or ethological description as a problem of prediction, in which observations are predicted by properties of fixed points of transforms of "pure systems", or by properties of convex combinations of such "pure systems". Other relationships to quantum methods are noted.Background 1 This paper follows and adopts background from [4, 5] which we summarize here. We assume a finite non-empty set P whose members are called individuals, and a finite non-empty set R whose members are called rules. An evolutionary structure is a quintuple S := (P, R, D, B, M) where D, B, and M are binary relations on P, satisfying these four axioms: (1) D is totally non-symmetric and transitive; (2) M is symmetric; (3) if bDc and there exists no d ∈ P, d = b, c for which bDd and dDc, then we write cP b, and require bBc iff for b, c, d ∈ P, dP b and dP c; (4) |bM| ≤ 2. A rule R ∈ R, R = ∅, is a statement concerning the relationships between the D, B, and/or M, which does not violate those four axioms. A family of subsets G = {G t |G t ⊆ P, t ∈ T} for t ∈ T a set of consecutive nonnegative integers starting with 0, is called a descent sequence of S. G t is called a generation of S, in case, for all G t ∈ G each cell bB occurs in only one generation, each subset bM occurs in only one generation, and for t > 0 when G t ∈ G, b ∈ G t , and cP b, then c ∈ G t−1 (that is, the set G t contains all of the immediate descendants of individuals in G t−1 ). We assume a "Darwinian Sequences axiom" which says that all descent sequences of a given evolutionary structure can be traced back through a chain of descent in an unbroken series of

Paths of Cultural Systems

“…Anthropologists very often draw illustrations of structures using methods discussed here, but based on intuition, thus have little notion of what their commonly used diagrams might predict. While [ 1 , 2 ] defined mathematical means to describe the current and future demographic organization of lineage organizations, with empirical examples, we here specify the demography of kinship-based systems [ 3 , 4 , 5 , 6 , 7 , 8 ] with some related definitions in our Appendix A ; and empirical examples in [ 9 , 10 , 11 , 12 , 13 ]. We follow the inspiration of [ 14 ].…”

“…Let H be a finite non-empty set of viable histories, let α ∈ H , let G t ∈ G be a generation of G, and let v ( t ) be the vector state of G t (see also Appendix A Definitions A2–A5). Then: From [ 2 , 5 , 6 , 8 ] and Appendix A Definition A4 each structural number s has a set of values n s and p s where n s p s = 2, where n s is the average family size of a pure system of structural number s and p s is the proportion of reproducing adults of a pure system of structural number s . If history α has structural number s , then each α has modal demography ( n α , p α ) = ( n s , p s ) (see Appendix A Definition A7) where p s = 2/ n s ; for s ≥ 3 and s α ≠ s χ then ( n α , p α ) ≠ ( n χ , p χ ).…”